Two dimensional neighborhoods of elliptic curves: analytic classification in the torsion case

We investigate the analytic classification of two dimensional neighborhoods of an elliptic curve with torsion normal bundle. We provide the complete analytic classification for those neighborhoods in the simplest formal class and we indicate how to generalize this construction to general torsion case.


Introduction and results
Let C be a smooth elliptic curve: C = C/ τ , where τ = Z + τ Z, with (τ ) > 0. Given an embedding ι : C → U of C into a smooth complex surface U, we would like to understand the germ (U, ι(C)) of neighborhood of ι(C) in U. Precisely, we will say that two embeddings ι, ι : C → U, U are (formally/analytically) equivalent if there is a (formal/analytic) isomorphism : (U, ι(C)) → (U , ι (C)) between germs of neighborhoods making commutative the following diagram (1.1) C ι id U C ι U By abuse of notation, we will still denote by C the image ι(C) of its embedding in U, and we will simply denote by (U, C) the germ of neighborhood.
However, we are still far, nowadays, to expect a complete description of the analytic classification in the non torsion case. It is the first case where the divergence between formal and analytic classification arises. Also, it is interesting to note that the study of neighborhoods of elliptic curves in the case deg(N C ) = 0 has strong reminiscence with the classification of germs of diffeomorphisms up to conjugacy. It will be more explicit later when describing the torsion case. The goal of this paper is to investigate the analytic classification when the normal bundle is torsion, and show that we can expect to provide a complete description of the moduli space in that case. More precisely, the formal classification of such neighborhoods has been achieved in [15]; we provide the analytic classification inside the simplest formal class, and we explain in Sections 10 and 11 how we think it should extend to all other formal classes, according to the same "principle". [22,33] in the case deg(N C ) = 0 and g > 0. There, among other results, he investigates the obstruction for the curve to be the fiber of a fibration (as it would be in the linear case (N C , 0) when N C is torsion). The Ueda type k ∈ Z >0 ∪ {∞} is the largest integer for which the aforementioned fibration 7 of N C can be extended to the k th infinitesimal neighborhood of C. In other words, k ≥ k 0 if the neighborhood can be covered by coordinate-charts (x i , y i ) : U i → C such that C ∩ U i = {y i = 0} and y i = a ij y j + o(y k 0 j ) on intersections U i ∩ U j with a m ij = 1 (m is torsion); see [3, section 2] for a more detailed exposition. When k = ∞, then we have a formal fibration, that can be proved to be analytic; the classification in that case goes back to the works of Kodaira, in particular in the elliptic case g = 1.

Formal classification. -An important formal invariant has been introduced by Ueda
Inspired by Ueda's approach, it has been proved by Claudon, Pereira and the two first named authors of this paper (see [3]) that a formal neighborhood (U, C) with deg(N C ) = 0 carries many regular (formal) foliations such that C is a compact leaf. This construction has been improved in [15,31] showing that one can choose two of these foliations in a canonical way and use them to produce a complete set of formal invariants. In the elliptic case g = 1, there are k m + 1 independent formal invariants for finite fixed Ueda type k where m is the torsion order N ⊗m C = O C (see [15]); for g > 1 and N C = O C (the trivial bundle), Thom founds infinitely many independent formal invariants in [31].
In this paper, we only consider the case g = 1 where N C is a torsion bundle. Most of the paper deals with the simplest formal class for which N C = O C is the trivial bundle, and Ueda type k = 1 is minimal. We explain in Sections 10 and 11 how to generalize to the more general case of torsion normal bundle N ⊗m C = O C , and finite Ueda type k < ∞. In the specific situation where k = 1, there are two scalar formal invariants ν, μ ∈ C. We associate a germ of neighborhood (U 1,ν,μ , C) as follows. Writing C as a quotient of C * by a contraction: C = C * z / < z → qz > with z = e 2iπx and q = e 2iπτ , (|q| < 1) we similarly define (U 1,ν,μ , C) as the quotient of the germ of neighborhood by the germ of diffeomorphism The two vector fields v 0 and v ∞ span a commutative Lie algebra, and therefore an infinitesimal C 2 -action on the quotient neighborhood. By duality, we have a 2-dimensional vector space of closed meromorphic 1-forms spanned by −ω 0 = dy y k+1 + ν dy y and ω ∞ = 1 2iπτ In particular, we get a pencil of foliations F t , t ∈ P 1 , by considering -either the phase portrait of the vector fields v t = tv 0 + v ∞ , -or ω t = 0 where ω t = ω 0 + tω ∞ .
When μ = 0, F ∞ defines a fibration transversal to the curve C and the neighborhood is the suspension 8 of a representation : π 1 (C) → Diff(C, 0) taking values into the oneparameter group generated by v 0 = y k+1 1+νy k ∂ y . For t ∈ C finite, F t is always (smooth) tangent to C, i.e. C is a compact leaf; when μ = 0, the same holds for F ∞ . For m + τ n ∈ \ {0}, F τ n m+τ n is the unique foliation of the pencil whose holonomy along the corresponding loop m + τ n in π 1 (C) ∼ is trivial (see Section 7). As proved in [15,Theorem 1.3], the neighborhoods (U 1,ν,μ , C) span all formal classes of neighborhoods with minimal Ueda class k = 1. For general k < ∞ with N C = O C , the scalar invariant μ is replaced by a polynomial P ∈ C[X] of degree < k, and the corresponding models (U k,ν,P , C) are described at the beginning of Section 10; the case of torsion N C is settled in Section 11. As explained in [15,Theorem 1.5], the moduli space of those neighborhoods with two convergent foliations in a given formal class up to analytic conjugacy is infinite dimensional, 9 comparable with C{X}. A contrario, if a third foliation is convergent, then the neighborhood is analytically equivalent to its formal model (U k,ν,P , C). However, an example of a neighborhood without convergent foliation is given by Mishustin in [21], and it is expected to be a generic property. In this paper, we describe the analytic classification 8 In the sense of foliations. 9 Isomorphic to Écalle-Voronin moduli spaces. of those neighborhoods formally conjugated to the simplest model (U 1,0,0 , C). As we shall see, the moduli space is comparable with C{X, Y}. We also provide in Sections 10 and 11 some evidence to the fact that a similar result holds more generally for torsion normal bundle N ⊗m C = O C , and finite Ueda type k < ∞.
There is an analytic isomorphism between complement of the curve S 0 \ C and C * × C * explicitly given by (1.2) : , ze 2iπτ ξ ). In this sense, we can view S 0 and P 1 X × P 1 Y ⊃ C * X × C * Y as two non algebraically equivalent compactifications of the same analytic variety. In fact, the algebraic structures of the two open sets are different as C * X × C * Y is affine, while S 0 \ C is not. Isomorphism (1.2) is therefore transcendental; we call it Serre isomorphism. Remark 1.1. -The complement of the curve S 0 \ C is known to be isomorphic to the moduli space of flat line bundles 10 over the elliptic curve, and has the structure of an affine bundle. The Riemann-Hilbert correspondence provides an analytic isomorphism with the space of characters Hom(π 1 (C), GL 1 (C)), which is isomorphic to C * × C * . This construction, due to Serre, provides an example of a Stein quasiprojective variety which is not affine (see [9, page 232], see also [32]). Precisely, it is shown that there is no non constant regular function on S 0 \ C. Logarithmic one-forms with poles supported on D correspond to the space of closed one-forms considered above via the isomorphism (see Fig. 1): Therefore, at the level of foliations, we have the following correspondence In particular, for m + τ n ∈ ∼ π 1 (C) in the lattice, the unique foliation with trivial holonomy along m + τ n corresponds to the one with rational first integral X m Y n : and the ruling corresponds to a foliation with transcendental leaves: Let us now study the isomorphism : S 0 \ C → P 1 × P 1 \ D near the compactifying divisors (compare [28]). Denote by V i a tubular neighborhood of L i in P 1 X × P 1 Y , of the form L i × disc say, and V = V 1 ∪ V 2 ∪ V 3 ∪ V 4 the corresponding neighborhood of D. Denote by V * = V \ D the complement of D in V. On may think of U 1,0,0 \ C as −1 (V * ). 11 Similarly, define V * i := V i \ D the complement of the divisor in V i and by U i = −1 (V * i ) the preimage: we have a decomposition neighborhood U 1,0,0 \ C = U 1 ∪ U 2 ∪ U 3 ∪ U 4 . One can show that U i s look like sectorial domains of opening π in the variable y saturated by variable z (see Section 3.1 and Fig. 2). Our main result is that this sectorial decomposition together with isomorphisms i : U i → V * i persists for general neighborhoods (U, C) in the formal class (U 1,0,0 , C); we conjecture and actually give the strategy to prove that a similar result holds true for all formal types, whenever N C is torsion (see Section 10). There is a formal isomorphism = z + m≥2 a m (z)y m , y + n≥4 b n (z)y n such thatˆ • F = F 1,0,0 •ˆ ; we have a m , b n ∈ O(C * z ) and no convergence assumption in y-variable. We can also considerˆ as a formal diffeomorphism (U, C) → (U 1,0,0 , C). The main ingredient of our classification result, proved in Section 9, is the (onto its image) havingˆ as asymptotic expansion 13 along C, satisfying After composition with the fundamental isomorphism : U 1,0,0 → C * X × C * Y , we get (see Fig. 3) After patching copies of germs 4 , together with a diffeomorphism germ which does not depend on the choice of sectorial normalisations i .
More generally, consider a neighborhood (V, D) in which each component L i ⊂ D has zero self-intersection. Then after [25], the neighborhood (V, L i ) is trivial (a product L i × disc). After identification with our model ψ i : we get that V takes the form V ϕ for a convenient 4-uple of diffeomorphisms ϕ. The gluing data ϕ is not unique as we can compose each embedding ψ i by an automorphism germ ϕ i ∈ Diff(V i , L i ). 15 Therefore, it is natural to introduce the following equivalence relation 13 The diffeomorphism i : U i → U 1,0,0 admitsˆ i as an asymptotic expansion along C if the entries of its lift i :Ũ i → C * z × C y admit the entries ofˆ as asymptotic expansion on each open subset C × S(I ε , r) (see Section 3.1). 14 These are germs of diffeomorphisms on V i ∩ V i+1 at p i,i+1 = L i ∩ L i+1 preserving restrictions of each L j . See Section 3.3. 15 These are germs of diffeomorphisms on V i along L i preserving restrictions of L i−1 and L i+1 . See Section 3.3.
Clearly, the moduli space V of neighborhoods (V, D) up to analytic equivalence 16 identifies with the set of equivalence classes for ∼. Notice that each equivalence class contains a representative ϕ such that ϕ 1,2 , ϕ 2,3 , ϕ 3,4 are tangent to the identity, and the linear part does not depend on the choice of such representative ϕ. Therefore, a, b ∈ C * are invariants for the equivalence relation, and we denote by V a,b the moduli space of those triples. With this in hand, we are able to prove: Theorem C.
-We have a one-to-one correspondence between Remark 1.2. -A thorough look to the proof of the Sectorial Normalization Lemma (9) may prove that the correspondence is analytic in the sense that analytic families of neighborhoods t → (U t , C) correspond to analytic families of cocycles t → ϕ t . As the freedom lies in the choice of (essentially) one-dimensional diffeomorphisms ϕ i , it is quite clear that the moduli space is essentially parametrized by two-dimensional diffeomorphisms, and therefore quite huge.
In a similar vein, it is reasonable to expect that the analytic moduli space U 1,ν,μ of neighborhoods (U, C) formally equivalent to (U 1,ν,μ , C) is in one to one correspondence with V a,b with a = e −4π 2 ν and b = e −4π 2 τ (ν+μ) . Actually, we explain in Section 10 how to construct an embedding V a,b → U 1,ν,μ , but the surjectivity needs to adapt our Sectorial Normalization Lemma. This creates additional issues (of purely technical nature) and we just indicate briefly how to proceed. Actually, one directly addresses in loc.cit the general case Ueda type = k, where we inherit 4k sectors with opening π k and the moduli space would be then equivalent to the moduli of neighborhoods of cycles of 4k rational curves (the model must be thought as a degree k cyclic étale cover of (V, D)). A precise statement, summarizing the structure of the analytic moduli space when N C O C , is given in Section 10, Theorem E. With this in hand, it is not difficult to undertake the analytic classification of neighborhood when N C is torsion. The idea consists in reducing to the case of trivial normal bundle by an appropriate cyclic cover. This is settled in Section 11.
Theorem D. -The foliationF t is convergent if, and only if, there exists a representative ϕ in the corresponding equivalence class such that each ϕ i,i+1 preserves the foliation In that case, these two foliations are conjugated via the isomorphism When F t is not of rational type, i.e. τ (1 − 1 t ) ∈ Q ∪ {∞}, then F t is defined by a closed meromorphic 1-form and the logarithmic 1-form of the statement is also preserved by all ϕ i,i+1 and defines a global logarithmic 1-form on (V, D). On the other hand, in the rational case, Écalle-Voronin moduli of the holonomy provide obstruction to define the foliation by a closed meromorphic 1-form. For instance, when F 0 is convergent, Martinet-Ramis cocycle are given by the X-coordinate of ϕ 1,2 • ϕ 2,3 and ϕ 3,4 • ϕ 4,1 (see Section 7.9 for details).
In [15], the two first authors with O. Thom provided the analytic classification of neighborhoods with 2 convergent foliations. In Section 7, we provide examples of neighborhoods with only one foliation, and also without foliation which is the generic case. An example without foliations has been given by Mishustin in [21] few years ago and it would be nice to understand what is the corresponding invariant ϕ.
In Section 8, we investigate the automorphism group of neighborhood germs. We prove in Theorem 8.1 that it can be of three types: finite (the generic case), one dimensional and we get an holomorphic vector field (and in particular a convergent foliation), or two dimensional only in the Serre example.
1.6. SL 2 (Z) action. -The analytic classification of resonant diffeomorphism germs of one variable is reminiscent in our classification result. However, there are strong differences like the fact that the sectorial trivialization is not unique in our case. Indeed, our sectorial decomposition U \ C = U 1 ∪ U 2 ∪ U 3 ∪ U 4 has been imposed by our choice of a basis for the lattice = Z + τ Z. It comes from the sectorial decomposition of the holonomy maps of the two foliations F 0 and F 1 having cyclic holonomy, trivial along 1 and τ respectively. If we change for another basis (m + τ n, m + τ n ), with m m n n ∈ SL 2 (Z) then the change of coordinates The new isomorphism is related to the previous one by a monomial transformation Using sectorial normalization for a general neighborhood (U, C) with this new basis gives a new compactification (V , D) which is bimeromorphically equivalent to (V, D).

1.7.
Concluding remarks. -Contrary to the diophantine case (non torsion normal bundle), the classification of neighborhoods of elliptic curves with torsion normal bundle can be completely described, as shown in Theorem E. A naive reading of Arnold's work [1] might suggest that classification of neighborhoods of elliptic curves with topologically trivial normal bundle could be similar to that of germs of one dimensional diffeomorphisms. In fact, the suspension of a representation π 1 (C) → Diff(C, 0) permits to embed the moduli space of diffeomorphisms into that of neighborhoods. However, this latter one turns out to be much more complicated, even if the general approach by sectorial normalization and classifying cocycle is still in the spirit of Écalle-Voronin classification for resonant diffeomorphisms, or Martinet-Ramis' version. For instance, an unexpected phenomenon in the case of neighborhoods is that the sectorial covering is not unique, due to the SL 2 (Z)-action. We can expect that the sectorial normalizations involve resurgent functions with lattice of singularities isomorphic to the lattice of the elliptic curve. Recall that the lattice of resurgence for resonant diffeomorphism has rank one. It would be interesting to better understand this phenomenon.
An important motivation to study neighborhoods was initially raised by Arnold: there is a close link with the study of germs of analytic diffeomorphisms of (C 2 , 0). Indeed, if we consider our model F 1,0,0 (z, y) = (qz, y 1−y ) at the neighborhood of (z, y) = (0, 0), then we get a semi-hyperbolic map whose space of orbits (when deleting z = 0) is obviously the neighborhood (U 1,0,0 , C) where C = C * / < qz >. One can investigate the analytic classification of small perturbations F := F 1,0,0 + · · · where dots are vanishing at sufficiently high order at the origin. Then, it is a classical fact that F has also an invariant manifold in the contracting direction |q| < 1, and the space of orbits gives rise to a neighborhood (U, C) formally equivalent to (U 1,0,0 , C). The analytic classification of these germs of semi-hyperbolic maps has been done by the last author with P. A. Fomina-Shaȋkhullina (see [27,38]) and comparing the two moduli shows that moduli of maps embed in moduli of neighborhoods but is infinite codimensional. In fact, the analytic extension of the map F to the origin imposes strong restrictions on the corresponding invariants ϕ defined in Section 1.4. It is interesting to consider the following hierarchy: 1. one dimensional resonant diffeomorphisms in (C, 0), 2. singular points of foliation in (C 2 , 0) of resonant-saddle or saddle-node type, 3. singular points of vector fields in (C 2 , 0) of resonant-saddle or saddle-node type, 4. singular points of diffeomorphisms in (C 2 , 0) of resonant-saddle or saddle-node (i.e. semi-hyperbolic) type, 5. neighborhoods of elliptic curves with torsion normal bundle.
The first occurrence gives rise to Écalle-Voronin moduli (see [4,5,36], and also [16]). One dimensional resonant diffeomorphisms also occur as monodromy map of those foliations arising in case (2). These latter ones have been classified by Martinet-Ramis in [17,18] and the classification on resonant-saddles and their monodromy map (1) turns out to be equivalent; however, saddle-node impose strong restriction to the invariants of its holonomy map (we can realize half of the moduli only). See [16][17][18] for details. Classification of vector fields has been done by the third author with Meshcheryakova (see [37] for instance) and independently by Teyssier [30]. This gives rise to twice the moduli space of foliations: the classification of vector fields with same underlying foliation is solved by the linearization of the conjugacy equation for foliations. Still, the moduli space is parametrized by finitely many copies of C{x} (power-series in one variable). There is a huge step when we pass to diffeomorphisms in (C 2 , 0) as the moduli space is now parametrized by copies of C{x, y}. Diffeomorphisms occurring in (4) are actually one-time-map of formal vector fields of type (3), but divergent as a rule. We expect that resonant-saddle diffeomorphisms (4) have same classification as neighborhoods (5), but classification of former ones looks somehow more delicate. It would also be nice to understand how Ueda's results [34,35] can be related to our work from that point of view. Also, diffeomorphisms of (C 2 , 0) arise as monodromy map of reduced singular foliations by curves in (C 3 , 0) and we can expect that the analytic classification is similar under generic conditions on the spectrum.
One might expect to investigate higher dimensional neighborhood of elliptic curve with trivial normal bundle by mimicking what has been done in [15] for the formal classification, and in the present paper regarding analytic classification. We haven't considered this direction at all. However, it might be interesting to note that Ueda's Theory has been generalized to higher codimension by Koike in [13]. What would be the higher dimensional analogue of Serre's isomorphism?

Preliminary remarks
Recall that C = C * / < q >, and we denote byC C * → C the corresponding cyclic cover. DenoteŨ = C * z × C y andC = {y = 0} ⊂Ũ. The following is already mentioned by Arnol'd [1]. Proof. -The self-intersection C · C determines topologically the germ of neighborhood. Then, by taking a suitable small representative, U is homeomorphic to a product D × C. So, one can consider the cyclic coveringŨ → U extending the cyclic cover C → C. This gives rise to a neighborhoodŨ ofC C * . Following Siu [29], the germ of this neighborhood alongC is isomorphic to the germ of a neighborhood of the zero section {y = 0} in the normal bundle NC C * z × C y . The deck transformation of the (germ of) covering takes the form F of the statement.
In that case, there exists a formal diffeomorphism (tangent to the identity along C) (2.4)ˆ (z, y) = z + n>0 a n (z)y n , y + n>1 b n (z)y n 19 Actually, one can, as usually, define analytic/formal conjugations between both neighborhoods in terms involving only the structural analytic/formal sheaves along C. The two definitions obviously coincide. with a n , b n ∈ O(C * z ) (and no convergence condition on y), such that Moreover, any other formal diffeomorphismˆ of the form (2.4) satisfying (2.5) writes Proof. -Let g = n∈Z g n z n be holomorphic on C * z . The functional equation admits a solution φ holomorphic on C * z if, and only if, g 0 = 0; then φ is unique up to the choice of φ(0). Indeed, if we write φ(z) = n∈Z φ n z n , then equation (2.7) writes φ n (q n − 1) = g n for all n.
Let f be a holomorphic non vanishing function on C * z . The functional equation admits a solution ϕ holomorphic and non vanishing on C * z if, and only if, -f : C * → C * has topological index 0 so that g = log(f ) is well-defined, -the coefficient g 0 of g = n∈Z g n z n vanishes.
Indeed, topological index is multiplicative and those of ϕ(qz) and ϕ(z) are equal and cancel each other. Then we can solve the corresponding equation (2.8) for g and set ϕ = exp(φ), which is unique up to a multiplicative constant. Note that, if g 0 = 0, then we can solve for a = exp(g 0 ).
Let us start with F like in (2.1). The change of coordinate 1 (z, y) = (z, ϕ(z)y) yields −1 We can easily check that the coefficient f in F defines the normal bundle N C in the quotient, and its topological index coincides with deg(N C ) which is zero in our case. Then we can find ϕ ∈ O * (C * z ) satisfying (2.9) and get Moreover, ϕ is unique up to a multiplicative constant. The coefficient a can be interpreted as a flat connection on N C with trivial monodromy along the loop 1 ∈ and monodromy a along the loop τ ∈ . In our case, N C = O C and a = 1 and we can write Solving equation (2.7), we get In our case, b = 0 (i.e. Ueda type k = 1). By using a change (z, λy) (freedom in the choice of ϕ above) we can set b = 1 and write The change of coordinate 3 Solving twice equation (2.7), we get F 3 (z, y) = (qz + αzy + O(y 2 ), y + y 2 + βy 3 + O(y 4 )).
Here, we have no freedom and α, β are formal invariants corresponding to μ, ν in the end of Section 1.4: in the formal class U 1,0,0 we get α = 0 and β = 1. Then, we can kill-out all higher order terms in F by a formal change of coordinate, or better normalize it to F 1,0,0 . Indeed, at the N th step, we get We can clearly normalize the two coefficients into brackets by a constant, and can even choose the constant by means of a, b.
The composition of all changes of coordinatesˆ −1 := 1 • 2 • 3 • · · · converges in the formal topology as a formal diffeomorphism satisfying (2.5). For any other formal diffeomorphismˆ of the form (2.4) satisfying (2.5), we have thatˆ :=ˆ •ˆ −1 is an automorphism of (U 0 , C) inducing the identity on C. As we shall see in Lemma 3.9,ˆ is necessarily convergent and of the form (2.6).

Sectorial decomposition and sectorial symmetries
In this section, we introduce the sectorial decomposition of U by transversely sectorial domains U i = −1 (V * i ) and compare spaces of functions on both sides. From now on, we work in the variable ξ = 1/y, at the neighborhood of ξ = ∞; this is much more convenient for computations. Notations are as in Section 1.3.
Note that P m (f ) is necessarily unique. Define Then one can associate to f its asymptotic expansion along {ξ = ∞}. This is a formal power serieŝ f ∈ O(C * z )[[ξ −1 ]] whose truncation at order m coincides with P m (f ). The asymptotic expansion is then unique, and we have a well-defined morphism of C-algebras The presheaf on S 1 defined by I → O I naturally gives rise to a sheaf on S 1 which we will denote by O. One can define on the same way the sheaves A m , A, A ∞ respectively associated to I → A m I , I → A I , I → A ∞ I and the last two are sheaves of differential algebras with respect to ∂ z and ∂ ξ . The stability by derivation is indeed a straightforward consequence of Cauchy's formula. As the asymptotic expansion is independent of the representative, we have a morphism of sheaves ] is viewed as a constant sheaf over S 1 ).

as a transversely sectorial open set of opening I i (in the sense of Definition 3.4). Moreover, the lift of
Proof. -For instance, for a, b, c > 0, we easily check that for 20 Mind that these intervals for arg(ξ ) correspond to those defined in Lemma A for arg(y) = − arg(ξ ).
The remaining cases are similar and straightforward. Denote Proof. -As before, we only give the proof for I 4,1 , the other cases are similar. If Conversely If we denote (X 0 , Y 0 ) = (z 0 , ξ 0 ), then the image of (z, ξ) while (s, t) runs over the square is a product of two loops. Therefore, the image (U 4,1 ) contains an open set W which is saturated by the toric action of , and which contains U 4,1 (just take W to be the image of all those (z, ξ) like above when (z 0 , ξ 0 ) runs over U 4,1 ). Since f is invariant under F 1,0,0 , i.e. f • F 1,0,0 = f , then it factors through and, maybe passing to another representative, we have f = g • where g ∈ O(W ). Mind that W (as well as (U 4,1 )) might not be of the form W \ (W ∩ D) for a neighborhood W of p 4,1 , but we will prove that the holomorphic hull of g is such a neighborhood.
As W is a Reinhardt domain, let us consider the (convergent) Laurent series of g: The coefficients are given by the integral . This can be rewritten as Now, given m, n ∈ Z, assume that there exists θ ∈ I 4,1 such that (e iθ (m + τ n)) < 0. The above inequality promptly implies that a m,n = 0 by fixing z 0 and making ξ 0 → ∞ in the direction θ (which is possible in U 4,1 as its opening is I 4,1 ). This is possible if, and only if arg(m + τ n) + I 4,1 intersects ] − π, 0[ mod 2π which, since I 4,1 =]0, π − [, means that It promptly follows that the only non zero coefficients a m,n occur when which means that m, n ≥ 0, and g extends holomorphically at p 4,1 : X = Y = 0. Finally, since f → 0 as ξ → ∞, we get that a 0,0 = 0 and g(0, 0) = 0.
Remark 3.8. -The second part of the proof does not use the fact that f is flat (i.e. admits asymptotic expansion zero) alongC, but only the fact that it is bounded. As a consequence, any bounded holomorphic function on a transversely sectorial open set U i or U i,i+1 as above automatically admits a constant as asymptotic expansion along C. We note that bounded functions on U 1 , U 3 (resp. U 2 , U 4 ) therefore correspond to first integrals of the foliation F 1 (resp. F 0 ).

3.3.
Sheaves of sectorial automorphisms. -Denote by Aut(S 0 ) the automorphism group of the ruled surface S 0 whose elements induce translations on C. It preserves the ruling as well as the section C ⊂ S 0 , inducing an action on the neighborhood of C. The subgroup Aut C (S 0 ) of elements fixing C point-wise is the one-parameter group generated by the flow of the vector field 21 We have an exact sequence where Aut 0 (C) is the translation group on C. The group Aut(S 0 ) is connected and generated by the flows of The full group of automorphisms Aut(S 0 ) must also preserve the ruling and C (this latter being the unique section of vanishing self-intersection). We claim that the restriction morphism is surjective and induces an isomorphism For a generic elliptic curve C, where Aut(C)/Aut 0 (C) has order 2, this is clear by considering the involution of S 0 (z, ξ) → ( 1 z , −ξ ). In full generality, one has Aut(C)/Aut 0 (C) Z/κZ where κ = 2, 3, 4 or 6. Fix κ and consider an associated lattice 1, τ ⊂ C defining C. One can present S 0 as the quotient C x × C ξ /G where G is the group generated by Let ω be a primitive κ th root of unit. Then, there exists a matrix M = m n m n ∈ In particular, h λ,μ lies in the normalizer of G, hence descend to S 0 , thus justifying the claim above.
In fact, specializing Aut(S 0 ) to the neighborhood of the curve, we get all analytic, and even formal automorphisms of the neighborhood (S 0 , C): Lemma 3.9. -Any formal automorphismˆ : (S 0 , C) fixing C point-wise is actually convergent and belongs to Aut C (S 0 ).
Proof. -Recall [15] that the only formal regular foliations on (S 0 , C) are those defined by ω = 0 where ω belongs to the vector space of closed 1-forms E = C dz z + Cdξ . Moreover, for ω ∈ E \ C dz z , F ω does not admit non constant formal meromorphic first integral, and the only formal closed meromorphic 1-forms defining F ω must be a constant multiple of ω, thus belonging to E. If is a formal automorphism of (S 0 , C) fixing C point-wise (b −1 non vanishing), then it must preserve the vector space E. In particular, it must preserve C dz z (and z actually as it fixes C point-wise) and sends dξ to some other element α dz z + βdξ . A straightforward computation shows thatˆ writeŝ (x, ξ) = z, α log(z) + βξ + γ , γ ∈ C, and we have α = 0. Finally, asˆ must commute with F 1,0,0 (z, ξ) = (qz, ξ − 1), we get β = 1. Proof. -The formal diffeomorphismˆ induces an automorphism of C. Using the surjectivity of the restriction morphism Aut(S 0 ) → Aut(C) and after composingˆ by a convenient element of Aut(S 0 ), we can assume that it fixes C point-wise, and then apply Lemma 3.9.
Definition 3.11. -Let us consider the germs of sectorial biholomorphisms in the direction arg(ξ ) = θ of (Ũ,C) that are tangent to the identity: The collection of these germs when varying θ naturally gives rise to a sheaf of groups (with respect to the composition law) on S 1 that will be denoted by G 1 . We will consider for further use the subsheaf G ∞ of G 1 of germs of sectorial biholomorphisms flat to identity, i.e. when f 1 , implies that its asymptotic expansion also commutes with F 1,0,0 , i.e.ˆ • F 1,0,0 = F 1,0,0 •ˆ . According to the description of the formal centralizer of F 1,0,0 in Lemma 3.9, it turns out that We would like to apply characterization of A ∞ [F 1,0,0 ](I) obtained in the previous section for our special sectors I i and I i,i+1 to obtain a similar characterization of sections ) the subgroup of germs tangent to the identity at p i,i+1 . For instance: Proposition 3.13. -We have the following characterizations: . Assume now I = I 4,1 , say. Then, by Proposition 3.7, one can write . Therefore, one can write Clearly, a, b are holomorphic at (X, Y) = (0, 0) and a(0, , by setting ) can be carried out exactly along the same line.

Analytic classification: an overview
Here, we would like to detail our main result, namely the analytic classification of all neighborhoods that are formally equivalent to (U 1,0,0 , C). The most technical ingredient is the sectorial normalization (Lemma A in the introduction) which now reads as follows. Let F be a biholomorphism like in Proposition 2.3 In particular, there is a formal diffeomorphismˆ (that can be assumed to be tangent to the identity along C) conjugating F to F 1,0,0 (z, ξ) = (qz, ξ − 1), i.e. F •ˆ =ˆ • F 1,0,0 . there is a section i of G 1 (I i ) (see Definition 3.11) such that Section 9 is devoted to the proof of this lemma. Let us see how to use it in order to provide a complete set of invariants for the neighborhood (U, First of all, we note that i is unique up to left-composition by a section of i.e. the composition of an element of the one-parameter group Remark 3.12). Using this freedom, we may assume that asymptotic expansions coincide: It follows that, on intersections I i,i+1 = I i ∩ I i+1 , we get sections Using Proposition 3.13, we have (see Fig. 4) In other words, setting i := • i , we get which proves Corollary B. We have therefore associated to each neighborhood (U, C) formally equivalent to (U 1,0,0 , C) a cocycle ϕ = (ϕ i,i+1 ) i∈Z 4 which is unique up to the freedom for the choice of i 's.
We will denote this equivalence relation by ≈.
Proof. -According to the description of Aut C (S 0 ), a biholomorphism germ (U, C) → (U , C) is indeed tangent to identity along C and then lifts-up to a global section ∈ G 1 (S 1 ) satisfying • F = F • . Let ( i ) and ( i ) be the sectorial normalizations used to compute the invariants ϕ and ϕ . Clearly, i • provides a new collection of sectorial trivializations for (U, C). We can write (using Remark 3.12) . After factorization through , using (3.3) and Proposition 3.13, we get the expected equivalence relation (4.3) for ϕ and ϕ . Conversely, if ϕ ≈ ϕ , then we can trace back the existence of an analytic conjugacy : (U, C) → (U , C) by reversing the above implications.
Remark 4.4. -We can weaken the notion of analytic equivalence between neighborhoods by considering biholomorphism germs : (U, C) → (U , C) inducing translations on C. This means that, in Definition 2.2, we now allow conjugacies (z, y) = (cz + O(y), O(y)) with c ∈ C * in formula (2.2), i.e. translations on the elliptic curve. In that case, the corresponding cocycles are related by 23 Having in mind the description given in (3.4), one observes that two cocycles are equivalent iff they lie on the same orbit over some action (that the reader will easily explicit) of the fiber product To summarize, we have just associated to each (U, C) and constructed a map from the moduli space U 1,0,0 of such neighborhood up to analytic equivalence an ∼ to the moduli space C of cocycles ϕ like (4.4) up to equivalence (4.3): which is proved to be injective in Proposition 4.3. In Section 5, we prove the surjectivity by constructing an inverse map ϕ → U ϕ . Before that, we want to reinterpret the cocycle ϕ as transition maps of an atlas for a neighborhood (V ϕ , D).

Construction of U ϕ
In this section, we construct a large class of non analytically equivalent neighborhoods, all of them formally equivalent to (U 1,0,0 , C). This is done by sectorial surgery, extending the complex structure along C by means of Newlander-Nirenberg Theorem. In order to do this, we have to work with smooth functions (i.e. of class C ∞ ).
Passing to inductive limits and sheafification as in Section 3.1, we get a sheaf E ∞ of differential algebra on the circle S 1 . Like in Section 3.3, we can also define the sheaf of groups D ∞ on the circle, whose sections ∈ D ∞ (I) are smooth sectorial diffeomorphisms asymptotic to the identity, i.e. of the form (z, ξ) The following property somehow expresses that a cocycle defined by a collection of sectorial biholomorphisms is a coboundary in the C ∞ category.
Proof. -One can extract from this covering a finite covering (J k ), k ∈ Z n such that only consecutive sectors J k and J k+1 intersect. It clearly suffices to prove the Lemma for this particular subcovering. Let (θ k ) a partition of the unity subordinate to this covering. Write One easily check that ψ k are smooth, equal to the identity outside intersections, and satisfy ψ k = k,k−1 • ψ k−1 on intersections as expected.
In particular, asymptotic Proof. -LetŨ i be the sectorial domain of definition of ψ i andŨ be their union together with the sectionC defined by ξ = ∞. Lemma 5.2 allows to write . In particular, denoting by I the standard complex structure on C 2 , J := ψ i * I = ψ j * I is a new complex structure onŨ \C which extends toŨ as a complex structure by Newlander-Nirenberg's Theorem. In fact, because of flatness of ψ i to the identity, the almost complex structure J extends at 0 as a C ∞ almost complex structure onŨ; by construction, it is integrable onŨ i 's and therefore Nijenhuis tensor vanishes identically onŨ \C, and by continuity onŨ. Then, Newlander-Nirenberg's Theorem tells us that J is integrable. Note that I = J in restriction toC which is then conformally equivalent to C * for both structures. Now, we use the fact that two-dimensional germs of neighborhood of C * are analytically equivalent as recalled in Section 2. This can be translated into the existence of a smooth diffeomorphism ψ of (Ũ,C) such that ψ * I = J. Up to making a right composition by a biholomorphism of (Ũ,C) with respect to I, one can suppose (exploiting that I = J on TŨ |C ) that ψ is tangent to the identity alongC. This implies that for every i, i := ψ i • ψ ∈ G 1 (U i ) and, because the ij 's are flat to identity, admit an asymptotic expansionˆ i alongC = C * independent of i. By construction, we Obviously, all along this proof, we might have shrinked the domainŨ of definition without mentioning it.
Remark 5.4. -The use of the Newlander-Nirenberg in this context is not new and can be traced back to Malgrange [16] and Martinet-Ramis [17].
We now specialize to our covering of S 1 determined by the intervals I i defined by (4.1) in Lemma 4.1. Let us show how to construct a neighborhood realizing a given . As i,i+1 commute to F 1,0,0 , we have on intersections: Therefore, we can define a global diffeomorphism of (Ũ,C) by setting on U i 's and extending by continuity as the identity mapping onC. By construction, the quotient has cocycle ϕ and is formally equivalent to U 1,0,0 . This proves the surjectivity of the map (4.5) whose injectivity has been proved in Proposition 4.3. It remains to prove the Sectorial Normalization Lemma 4.1 (i.e. Lemma A in the introduction), which will be done in Section 9. Modulo this technical but central Lemma, we have achieved the proof of Theorem C.

Construction of V ϕ
In this section, keeping notations of Section 1.3, we generalize Serre isomorphism : has trivial normal bundle, and an isomorphism germ canonically attached to the analytic class of (U ϕ , C) in the following sense: if (U ϕ , C) is another neighborhood germ, then where the analytic equivalence allows translations 24 on C for the left-hand-side, and preserves the numbering of lines L i on the right-hand-side.
Proof. -Given a cocycle not necessarily tangent to the identity we define a new germ of analytic neighborhood of D as follows. We consider the disjoint union of neighborhood germs (V i , L i ), and patch them together through the transition maps The resulting analytic manifold V ϕ contains a copy of D, namely the union of lines L i identified at points p i,i+1 , and only the germ of neighborhood V ϕ makes sense This germ of neighborhood comes with embeddings Conversely, if (V, D) is a germ of neighborhood of D where all lines L i have zero selfintersection, then there exist trivialization maps (preserving D) 24 We emphasize that this is not exactly the equivalence relation defined in Definition 2.2.
(where (V, L i ) denotes the germ of V along L i ) in such a way that, near p i,i+1 we have It is clear from above arguments that, for another cocycle ϕ , we have and in that case, the isomorphism V ϕ ∼ −→ V ϕ is given by patching From the linear part of equivalence relation ϕ ∼ ϕ , we see that any cocycle ϕ is equivalent to a cocycle such that The pair (a, b) is an invariant of the neighborhood V ϕ . Cocycles arising from (U, C) for ∼ (U 1,0,0 , C) have invariants a = b = 1. In order to prove the equivalence (6.2), we just have to note that, for equivalent cocycles ϕ ∼ ϕ normalized as above (in particular when all ϕ i,i+1 , ϕ i,i+1 are tangent to the identity) then all four conjugating maps ϕ i have the same linear part. Then apply Remark 4.4 to show that it corresponds to analytic equivalence of (U ϕ , C) and (U ϕ , C) up to a translation of the curve.
Finally, we construct the isomorphism (6.1) by patching together the sectorial ones

Foliations
Recall that our model (U 1,0,0 , C) carries a pencil of foliations moreover, there is no other formal foliation on (U 1,0,0 , C) either tangent, or transversal to C (see [15,Section 2.3]). Via the isomorphism : U 1,0,0 \ C → V 0 \ D, we get the where one recalls that (X = e 2iπξ , Y = ze 2iπτ ξ ). The monodromy (or holonomy) of F t is given by In particular, for m n ∈ Q ∪ {∞}, are equivalent -F t has trivial monodromy along m +τ n ∈ \{0}, viewed as a loop of π 1 (C) ; -t = τ n m+τ n , or equivalently 1 t − 1 τ = m n ; - * F t admits the rational first integral X m Y n .
We will say that F t is of rational type if there is m n ∈ Q ∪ {∞} with these properties, and of irrational type if not. We note that rational type foliations are characterized by the fact that their holonomy group is cyclic (one generator), and also that the space of leaves (after deleting C and regarding them as global foliations on S 0 ) is somehow "rational", and not "elliptic".
If (U, C) is any analytic neighborhood with a formal conjugacŷ then it also carries the pencil of formal foliationsF t :=ˆ * F t . As we shall prove, these foliations are divergent in general. In fact, recall (see [15,Theorem 4

])
Theorem 7.1. -Let (U, C) be an analytic neighborhood formally equivalent to (U 1,0,0 , C). Assume three elementsF t 1 ,F t 2 ,F t 3 of the pencil are convergent, -or two elementsF t 1 ,F t 2 of the pencil are convergent, both of irrational type: 1 Then the full pencilF t is convergent, and (U, C) is analytically equivalent to (U 1,0,0 , C) (in factˆ is convergent).
In [15,Theorem 5], the two first authors and O. Thom construct infinite dimensional deformations of neighborhoods with two convergent foliationsF t 1 andF t 2 , provided that one or two of them is of rational type. In fact, Écalle-Voronin moduli spaces are shown to embed in moduli spaces of neighborhoods through these bifoliated constructions. Now, we know from our main result Theorem C, that the moduli space of neighborhoods is larger, comparable with C{X, Y}, in contrast with Écalle-Voronin moduli space which is comparable with C{X}. The point is that we have missed all neighborhoods with only one, or with no convergent foliation in the aforementioned work.
If the formal foliationF t of (U, C) is convergent, then the induced foliation i * Ft on V * i extends as a singular foliation on V i and is defined by a closed logarithmic 1-form We note that are equivalent: θ is a closed logarithmic 1-form on V i with non trivial poles supported on D ∩ V i , θ = α dX X + β dY Y + η with (α, β) = (0, 0) and η holomorphic and closed on V i , For instance, on V 1 , if η = df with f ∈ C{Y}, f (0) = 0 and α = 0, one can choose Proof. -Let us start with the tangent case t = ∞. By transversality of F 0 and F ∞ , and the fact that the ω t 's are F 1,0,0 -invariant, one deduces that the sectorial foliation i * Ft on (U 1,0,0 , C) is defined by a unique 1-form writing as 0) and we arrive at a similar situation Of course, we have used t = 0, 1 in order to get non zero multiple of ω. Let us end with the caseF ∞ . The foliation, in that case, can be defined by a closed holomorphic 1-form ω extending the holomorphic 1-form on C. Now, up to a multiplicative constant, we can write i * ω = ω ∞ + η for a closed 1-form η = uω 0 + vω ∞ where u, v ∈ A ∞ [F 1,0,0 ](I i ) One concludes as above.
Lemma 7.4. -If the formal foliationF 0 (resp.F 1 ) of (U, C) is convergent, then the induced foliation i * Ft on V * i extends as a singular foliation on V i and is defined by a 1-form Again, by Proposition 3.7, we see that if i = 1, 3, then u = f i (Y) with f i as in the statement, and we are done. However, when i = 2, 4, then u = f (X), but we cannot divide by f 1−f to get a closed logarithmic 1-form as before: the multiplicity of the polar locus is likely to increase (see remark 7.5).
Remark 7.5. -In Lemma 7.4, we can always define the foliation i * Ft by a closed meromorphic 1-form on V i provided that we allow non logarithmic poles. For instance, in the case t = 0 and i = 2, 4, if f ≡ 0 (is identically vanishing), there is nothing to do, it is the logarithmic case; if f ≡ 0, then after division, we get withf ∈ C{X},f (0) = 0, and k ∈ Z >0 . As it is well-known (see [15, Section 2.2]), we can write for some α ∈ C (the residue off (X) dX X k+1 ) and ϕ i ∈ Diff 1 (V i , L i ).
We can now prove Theorem D.
Corollary 7.6. -The formal foliationF t of (U ϕ , C) is convergent if, and only if, there exist Equivalently, there exists an equivalent cocycle ϕ ∼ ϕ such that Proof. -When t = 0, 1, the proof easily follows from Lemma 7.2. Indeed, all i * Ft are defined by θ i = 0 and have to patch via the glueing maps ϕ i,i+1 . Conversely, if θ i = 0 patches via the glueing maps ϕ i,i+1 , then this means that we get a foliation F on U ϕ \ C which is flat toF t along C, and therefore extends by Riemann. Using Remark 7.3 and Definition 4.2, one easily derives the second (equivalent) assertion. Finally, in the case t = 0 for instance, after applying Lemma 7.4 in a very similar way, we note that θ i defines a regular foliation on V i for i = 1, 3 (as η i = df , f ∈ C{Y ±1 }). On the other hand, on V i for i = 2, 4, θ i defines a singular foliation as soon as f i ≡ 0 (non identically vanishing), i.e. with a saddle-node singular points at the two points p i,i+1 and p i−1,i ; therefore, f i ≡ 0 in the case we have a global foliation and we are back to the logarithmic case. The proof ends-up like before.
Remark 7.7. -The statement of Corollary 7.6 can be reformulated as follows. The formal foliationF t of (U ϕ , C) is convergent if, and only if, there exists a foliation G t on (V ϕ , D) which is locally defined by a closed logarithmic 1-form with poles supported on D and having residues t on L 1 and (1 − t)τ on L 4 (we have automatically opposite residues on opposite sides of D). Indeed, the local foliations θ i = 0 patch together.
We can precise Corollary 7.6 for generic t as follows. Obviously, ω = * ϕ θ up to a constant.
Proof. -WhenF t is not of rational type, then we have and f i is a meromorphic first integral for θ i = 0, which must be constant in the irrational type. This constant must be = 1 as the residues are preserved. As a consequence, all θ i patch together on V ϕ . Finally, note that ifF t is convergent, then its holonomy is not cyclic (because not of rational type) and therefore preserves a meromorphic 1-form on the transversal that we can extend as a closed meromorphic 1-form ω defining the foliation.
Let us now illustrate how different is the situation for foliations of rational type by revisiting the classification [15, Theorem 5] of neighborhoods with 2 convergent foliations, in the particular case ofF 0 andF 1 , corresponding respectively to vertical and horizontal foliations on V ϕ . The proof is a straightforward application of the above criteria.
for 1-variable diffeomorphisms α i tangent to the identity, and the corresponding foliations on V ϕ are respectively defined in charts V i by dX = 0 and dY = 0. Moreover, this normalization is unique up to conjugacy by φ(X, Y) = (e t X, e τ t Y).
The space of leaves ofF 0 on U \ C corresponds to the space of orbits for its holonomy map, and therefore to Martinet-Ramis' "Chapelet de sphères" (see [18, page 591]). It is given by two copies of C * X patched together by means of diffeomorphism germs α 1 (X) at X = ∞ and α 3 (X) at X = 0. A similar description holds forF 1 with Martinet-Ramis' cocycle α 2 and α 4 . The invariants found by the third author in [36] are related with the corresponding periodic transformations in variable ξ (see Fig. 5).
Remark 7.10. -It follows from [15], or from the unicity of the formal pencilF t , that for given t ∈ P 1 , one cannot find two different collections (θ i ) i and (θ i ) i defining two global logarithmic foliations G t and G t on V, like in Corollary 7.6. One way to see this directly from the point of view of this section is as follows. In the irrational case 1 t − 1 τ ∈ Q, we see from Proposition 7.8 that θ i 's patch as a global closed logarithmic 1-form. But the difference between two closed logarithmic 1-forms with the same residues is a closed holomorphic 1-form η on (V, C). Now, η must be zero, even if we restrict on two consecutive line neighborhoods , as it only depends on X or on Y depending on the sector. In the rational case, 1 t − 1 τ ∈ Q, there is also unicity of θ i 's on two consecutive line neighborhoods whose residues have quotient > 0; indeed, after blowing-up, we get a rational fibration which must be unique by Blanchard Lemma.

Non existence of foliations. -For a generic neighborhood (U ϕ , C)
, there is no convergent foliation. In order to prove this, it is enough to provide a single example without foliation. Such an example has been given quite recently by Mishustin in [21]. With our Corollary 7.6, it is not too difficult to provide an example without foliations.
Theorem 7.11. -Let (U ϕ , C) be a neighborhood such that Then all foliationsF t belonging to the formal pencil (F t ) t∈P 1 are divergent.
Even the transversal fibrationF ∞ is divergent in that case.

7.3.
Only one convergent foliation. -To complete the picture, it is interesting to provide an example of a foliation having only one convergent foliationF t 0 in the pencil (F t ) t∈P 1 for an arbitrary t 0 .
Theorem 7.12. -Let t 0 = [u 0 : v 0 ] ∈ P 1 , and let (U ϕ , C) be the neighborhood such that ThenF t 0 is the unique convergent foliation in the formal pencil (F t ) t∈P 1 .
Proof. -Note that ϕ preserves the foliation defined by the logarithmic form α 0 dX X + β 0 dY Y which then descends on V ϕ , i.e. the formal foliationF t 0 is indeed convergent on U ϕ . Assuming by contradiction that there is another convergent foliationF t with t = t 0 , let θ i be the associated logarithmic 1-form on V i . As in the proof of Theorem 7.11, we get The residual parts of the 2-form θ 4 ∧ (ϕ 1,4 ) * θ 1 at X = 0 and Y = 0 respectively write We are led to a similar discussion as in the proof of Theorem 7.11. When β = 0, then f 1 ≡ 0 and (ϕ 1,4 ) * θ 1 only depends on X(1 + XY) −β 0 , while θ 4 only depends on X; we get a contradiction since β 0 = 0 in this case. When α = 0, then f 4 ≡ 0 and (ϕ 1,4 ) * θ 1 only depends on Y(1 + XY) α 0 , while θ 4 only depends on Y; contradiction. Finally, when both α = 0 and β = 0, then which cannot be zero, again a contradiction.
Remark 7.13. -More generally, given f ∈ C{X, Y} vanishing along X = 0 and Y = 0, the same proof shows that the cocycle defined by ϕ 1,4 = (Xe −β 0 f , Ye α 0 f ) and ϕ i,i+1 = Id otherwise also provides a neighborhood (U ϕ , C) with only one convergent foliation, namelyF t 0 , provided that df ∧ d(X p Y q ) ≡ 0 for all p, q ∈ Z >0 . Moreover, one easily checks that two different such f , says f and f , define non equivalent neighborhoods provided that their difference do not take the form f − f = g(X) + h(Y).

Symmetries
We follow notations of Sections 3 and 4. Let (U ϕ , C) be a neighborhood formally equivalent to (U 1,0,0 , C). Via formal conjugation, formal symmetries (or automorphisms) of (U ϕ , C) which restrict to translations on C are those of (U 1,0,0 , C). They form the group denoted Aut(S 0 ) and are explicited in the left hand side in the correspondence below (see Corollary 3.10): The subgroup Aut(U ϕ , C) of convergent automorphisms (which restrict to translations on C) thus identifies with a subgroup of the two dimensional linear algebraic torus: Theorem 8.1.
-Let (U ϕ , C) and G be as above. Then the subgroup G ⊂ C * × C * is algebraic. In particular, we are in one of the following cases: -G is finite and G = {(a, b) ; a p b q = a p b q = 1} for some non proportional (p, q), (p , q ) ∈ Z 2 \ (0, 0); -G = {(a, b) ; a p b q = 1} for some (p, q) ∈ Z 2 \ (0, 0); in particular, a finite index subgroup of G is generated by the flow of the rational vector field pX∂ X + qY∂ Y ; -G = C * × C * and (U ϕ , C) an ∼ (U 1,0,0 , C).

Moreover, in the first two cases, up to equivalence ≈, the cocycle takes the form
where u i,i+1 , v i,i+1 are Laurent series in X p Y q and X p Y q in the first case, in X p Y q in the second one and the action of G is linear in each chart (V i , L i ).
According to the correspondence depicted in (8.1) and Proposition 3.13, there exists a collection of automorphisms Moreover, these g i s define an automorphism of the neighborhood of four lines (V ϕ , D) associated to U ϕ . The gluing conditions read Note that the corresponding element (a, b) to h in C * × C * is the linear part of g i at the crossing points p i−1,i , p i,i+1 and is independent of i.
Conversely, the datum of (g i ) ∈ i Diff(V i , L i ) fulfilling (8.2) determine f (use the necessary and sufficient condition provided by Proposition 3.13).
We claim that g i can be linearized in each chart (V i , L i ). Let us consider for instance the transformation g 4 on (V 4 , L 4 ). Let us first prove that g 4 is locally linearizable at intersection point p 4,1 . We note that g 4 preserves the vertical fibration dX = 0 and sections Y = 0, ∞; it therefore takes the form The gluing condition involving ϕ 4,1 shows that the restriction g 4 | L 1 : X → aX · u(X) becomes linear in the chart (V 1 , L 1 ). Therefore, after changing the X coordinate on (V 4 , L 4 ), we can assume On the other hand, g 1 preserves the fibration dY = 0 of (V 1 , L 1 ), and by compatibility condition, g 4 must also preserve the local foliation with first integralȲ = ϕ * 4,1 Y near p 4,1 . In other words, in new coordinates (X,Ȳ), the transformation g 4 preserves dX = dȲ = 0 and is linear. Now, in order to prove the claim, we have to show that this linearization can be done globally along (V 4 , L 4 ). For this, consider the change of coordinateȲ = φ(X, Y) used to linearize locally and decompose φ(X, Y) = k≥1 φ k (X)Y k . Expanding the linearization equation in powers of Y shows that the truncated change of coordinatẽ Y = φ 1 (X)Y also linearizes g 4 .
As a by-product, G must contain the Zariski closure of < (a, b) >⊂ C * × C * . Indeed, all ϕ i,i+1 have to commute with g(X, Y) = (aX, bY). Writing ϕ i,i+1 (X, Y) = (X · u(X, Y), Y · v(X, Y)), we see that u, v have to be invariant by g, i.e. u • g = u for instance; equivalently, all non zero monomials of u and v are g-invariant. These monomials define an algebraic subgroup H ⊂ C * × C * which is the group of linear transformations commuting with ϕ i,i+1 . We conclude that (a, b) ∈ H ⊂ G.
If g was Zariski dense in C * × C * , we are done: the commutation of ϕ i,i+1 with all linear transformations shows that ϕ i,i+1 is linear (hence trivial) and (U ϕ , C) an ∼ (U 1,0,0 , C). Now, assuming that G is a strict subgroup of C * × C * , we want to prove that it can be linearized globally.
Assume first that G is finite. Then it can be linearized on each line neighborhood (V i , L i ). Indeed, for instance on (V 4 , L 4 ), G acts by transformations of the form and we denote by lin(g) its linear part (aX, bY). Then the transformation , u(0) = v(0) = 1 and linearizing the group: We can therefore assume that G acts linearly in each chart (V i , L i ) and the cocycle ϕ has to commute with all elements. It is well known that the group G is generated by two elements (a 1 , b 1 ) and (a 2 , b 2 ) of finite order; moreover, by duality, G is defined by 2 independent monomial equations a p b q = a p b q = 1. Gluing conditions with ϕ i,i+1 show that u i,i+1 , v i,i+1 must be G-right-invariant and therefore factor through the two monomial equations.
On the other hand, if G contains an element g of infinite order, then we can first linearize this element. The Zariski closure H of its iterates < g > in C * × C * is one dimensional, if strictly smaller than C * × C * , and defined by a monomial equation a p b q = 0. If G is larger than H, then the quotient G/H is generated by an element of finite order g ∈ G, and we can linearize the finite group < g > like above; since g and its linear part both commute with H, the linearizing transformations also commute with H and G is linearized. It is therefore algebraic, defined by monomial equations of ϕ i,i+1 , and they all factor into a single monomial. Remark 8.3. -From the description above, we note that the convergence of a non trivial automorphism g of (U ϕ , C) inducing the identity on C implies that (U ϕ , C) an ∼ (U 1,0,0 , C), since the group generated by g must be Zariski dense in C * × C * .

Sectorial normalization
We maintain the foregoing notations. Recall that we have set = arg τ ∈]0, π[. Let (U, C) be formally equivalent to (U 1,0,0 , C). We want to show that (U, C) has the form U ϕ . In other word, we want to prove Lemma A, or equivalently Lemma 4.1.

Overview of the proofs. -One can suppose that
Here, the terms i i = 1, 2 appearing in the residual part are well defined on a neighborhood of {ξ = ∞} in C * ×C}. Moreover, one can require that i = O(ξ −N ) 25 25 It must be understood that for every compact K of C * , there exists C K > 0 and M K > 0 such that for all (z, ξ) such that z ∈ K and |ξ | ≥ M K , where N 0 is an arbitrarily large integer (depending on the order of truncation of the formal conjugacy map), so that there exists a formal diffeomorphism (z, ξ) = (z +ĝ, ξ +ĥ),ĝ = n≥1 b n ξ −n ,ĥ = n≥1 a n ξ −n where a n , b n are entire functions onC = C * such that This can be reformulated as Basically, we will show that there exists a holomorphic solution = Id + (g, h) of the previous fonctional equations F • = • F 1,0,0 , i.e. with h, g satisfying defined on "suitable sectorial domains", namely on U i := −1 (V i −L i ), i = 1, 2, 3, 4, and admitting asymptotic expansion alongC compatible with the formal conjugacy mapˆ . More precisely g, h ∈ A(I i ) with the notations of section 3.1 with respective asymptotic expansionsĝ andĥ. To this end, we will first exhibit solutions in O(I i ) satisfying some suitable growth behaviour of the following linearized equations: Most of Section 9 is devoted to the construction of such sectorial solutions on the sector U 1 , and it will be explained in Section 9.6 how to deduce normalization on other sectors. This will enable us to solve by a fairly standard fixed point method the initial functional equations (9.3) and (9.4). In order to get rid of the coefficient q on the left hand side, note that both equations (9.4) and (9.3) can be reformulated as: (1 +g(z, ξ )), ξ + h(z, ξ) where the symbol stands for the same modification than in Remark 9.1 for the linear case.

The linearized/homological equation. -
Our purpose is to construct some sectorial solution of the linearized functional equations (9.6) (and therefore (9.5) by Remark 9.1) belonging to O(I 1 ). Actually, one will just firstly state some results and use this material to undertake the resolution of the complete (non linear) conjugacy equation (over I 1 ). We will detail the resolution of the linearized equation (the most technical part) in Section 9.5. The existence of other sectorial conjugacy maps over I i , i = 1 can be obtained in a very similar way and we indicate briefly how to proceed in Section 9.6.
In what follows, we will indeed provide a solution of (9.6) with "good estimates" on a domain of the form S |q|a,b,δ using a "leafwise" resolution with respect to the foliation defined by the levels of Y. For the sake of notational simplicity we will omit for a while the subscript a, b by setting S δ := S a,b,δ and S δ := S |q|a,b,δ . If δ 1 ≤ δ 2 , remark that S δ 1 and S δ 1 are respectively subdomains of S δ 2 and S δ 2 . To state precisely our result, let us fix some additional notations and definitions. Let m ≥ 3 a positive integer and consider the subspace H m δ of H(S δ ) 26 defined by the functions such that We will also introduce the space H ∞ δ of bounded holomorphic functions h on S δ equipped with the natural norm Theorem 9.2. -Fix a, b as above. Then, there are positive constants δ (a,b) , C such that: for every δ ≤ δ (a,b) and every function δ ∈ H m δ , m ≥ 3, there exists a unique function h δ ∈ H ∞ δ satisfying In addition, there exists a positive constant D θ only depending on θ ∈]0, π 2 ] such that As mentioned before, we will postpone the proof of Theorem 9.2 to Section 9.5. Condition (3) is needed for the unicity, and after to produce a norm with a unique fixed point when solving the functional equation. For the time being, we detail how it provides a section 1 of G 1 (I 1 ) of the form (z + g, ξ + h) such that the pair (g, h) admits (ĝ,ĥ) as asymptotic expansion and satisfies in addition the equations (9.8) and (9.7). In other words, we are going to exhibit a transversely sectorial conjugacy map between F and F 1,0,0 :

9.3.
Solving the functional equation. -Notations as in Theorem 9.2. It is worth mentioning that the strategy developed here as well as the resolution of the linearized/homological equation in the forthcoming Section 9.5 owes a lot to [38].
As before, a, b are fixed, δ (a,b) > 0 is small enough and may be adjusted from line to line in order to guarantee the validity of the estimates below. We will denote by δ any positive number such that 0 < δ ≤ δ (a,b) . We will omit for a while the subscript (a, b). Let us introduce two Banach spaces.
Note that both normed spaces are Banach spaces, and from Theorem 9.2, one inherits a continuous linear map between them: defined by solving To be more precise, for every δ small enough, and every with the positive constant C given by Theorem 9.2. We now define a non linear continuous map in the other way. Let us come back to the expression of the transformation F = F 1,0,0 + ( 1 , 2 ) defining the formally equivalent neighborhood (U, C) as explicited in Section 9.1. One can assume i (z, ξ) = O(ξ −N ) for a fixed arbitrary integer N ≥ 4. Recall that the i 's are analytic on a neighborhood of {ξ = ∞} and consequently are well defined as an element of H m δ , m < N, whose m norm tends to zero when δ goes to zero. For every M > 0, let us denote by the respective closed balls of radius M. Then, for δ small enough, we a have a well defined map In particular, R is continuous (Lipschitz).
Let ε > 0 such that εC < 1. Then, the composition L • R induces a (non linear) contracting map of the complete metric space H ∞,∞ δ (1). The unique fixed point is a solution to the functional equations (9.8), (9.7). This provides a solution of the original functional equations (9.6) and (9.5), taking into account the renormalization indicated in Remark 9.1. By uniqueness, the solution − → h δ attached to δ induces by restriction the solution attached to δ for δ ≤ δ.
One can complete this picture by taking into account all the properties required in the statement of Theorem 9.2. This leads to the following list of properties of the solution exhibited above as a fixed point of a non linear operator. We reintroduce the subscript (a, b) (with obvious notations) in order to recall that the choice of δ depends on a fixed arbitrary annulus in the z variable:  ≤ δ (a,b) , the system of equations (9.7), (9.8) admits a unique solution (g δ,a,b , h δ,a, a,b , h δ ,a,b ) is the restriction of (g δ,a,b , h δ,a,b ) a,b , h δ,a,b ) = 0.
there exists a positive number D = D(θ ) depending only on θ ∈]0, π 2 ] such that for every δ ≤ δ (a,b) and every (z, ξ) 9.4. Asymptotic expansion. -Notations as above. We start by fixing N ≥ 4 and a, b as before. Let us denote by (g a,b , h a,b ) the germ of sectorial solution induced by (g δ,a,b , h δ,a,b ) by taking δ → 0. Let a , b be positive real numbers such that a ≤ a < b ≤ b . By Proposition 9.4, note that the unique solution of (9.7), (9.8) In particular, (g δ,a,b , h δ,a,b ) is the restriction of (g δ,a ,b , h δ,a ,b ). Then, if one takes projective limit with respect to a → 0, b → +∞ and exploits the last asymptotic estimate in the Proposition 9.4, one get a solution (g, h) of (9.7) and (9.8) well defined as a flat element of A m−3 (I 1 ) × A m−3 (I 1 ). Set = Id + (g, h) with g = zg so that, by virtue of 9.1, where equality takes place in O(I 1 ) componentwise. Now, consider an integer p N arbitrarily large. Let k ≥ p be a positive integer and consider the truncation (or k-jet) J kˆ ofˆ at order k: If k is large enough, then one has can apply Proposition 9.4 to get existence and uniqueness ofg k where g k δ (z, ξ) = zg k δ (z, ξ). As before, one can invoke uniqueness and restrictions considerations to infer that these solutions are in fact induced by a flat element of A p−4 (I 1 ) × A p−4 (I 1 ) and that = J kˆ • k . Thus, is flat componentwise. As p can be chosen arbitrarily large, we eventually get that ∈ G 1 (I 1 ) and admitsˆ as asymptotic expansion. We have thus obtain the sought normalization 1 := on the germ of sector of opening I 1 . Note that 0<|c|<δ S a,b,c = S a,b,δ and the linearized equation has a simple form restricted to these slices. To simplify the presentation, we introduce the notation (see Fig. 6) ζ = 2iπτ ξ and λ := −2iπτ with λ = λ 1 + iλ 2 , λ 1 , λ 2 ∈ R and note that the real part λ 1 > 0. In particular, the linearized/homological equation can be then rewritten as where ϕ c (ζ ) = h δ (ce −ζ , − ζ λ ), and c (ζ ) = δ (ce −ζ , − ζ λ ). We are then led to solve the family of difference equations (9.9) with respect to the parameter c in the vertical strip where we impose ϕ c to be holomorphic, defined on the larger strip |q|a,b,c = {ζ ∈ C : log |c| − log b ≤ (ζ ) ≤ log |c| − log a + λ 1 } and to depend analytically on the parameter c in order to recover a holomorphic solution to (1) in Theorem 9.2.

Resolution of a difference equation. -
We now proceed to the construction of ϕ c . It is essentially a consequence of the following general result (with additional estimates).

and let be holomorphic on the strip
Suppose moreover that, for some m ≥ 3, we have: Then, there exists a bounded holomorphic function ϕ on A,B+λ 1 which solves Moreover ϕ is unique modulo an additive constant.
Concerning the existence part, let us first observe, by Cauchy formula, that where L − = { t = A} and L + = { t = B} are both oriented from bottom to top. Since m < ∞, we see that the two integrals are well defined, and holomorphic in ζ . Observe that F − 0 and F + 0 are respectively defined on the half-planes A < ζ and ζ < B, and can be extended to the boundary by continuity, likely as , by using equality (9.11). Then we define, for n ≥ 0 The solution ϕ to (9.10) is therefore given by the following series that will be proved to converge uniformly on the large strip A,B+λ 1 in Lemma 9.6. We can already check that it is indeed a solution: . Therefore, Theorem 9.5 is an immediate consequence of the following Lemma which actually provides further informations.
and then by reversing the role of L − and L + in the previous estimate, Following the same principle, we get the inequality This eventually leads to the upper bound (9.15) |F n (ζ )| ≤ 2C 1 m I m ∀n ≥ 0, ∀ζ ∈ A,B+λ 1 .
Moreover, when n > B−A λ 1 , Cauchy's formula allows to write Consequently, for these values of n one has The key point will be to find a suitable upper bound for where n 0 > 2 B−A λ 1 + 1 , t ∈ L + , ζ ∈ A,B+λ 1 . For n ≥ n 0 , Cauchy-Schwarz inequality gives where v = (t − ζ ) and λ = λ 1 + iλ 2 . Here, we have used the fact that Therefore, This, together with the fact that m < ∞ for some m ≥ 3, proves immediately that the serie F n converges uniformly on every compact of A,B+λ 1 . The function ϕ is then well defined and holomorphic on A,B+λ 1 .
For n ≥ n 0 , consider the decomposition F n = G n + H n where Consequently, when |v| ≤ 1, we deduce that |t − ζ | ≤ √ 1 + (B − A) 2 so that, by (9.17), we find n≥n 0 |G n (ζ )| ≤ C 2 m I m where In order to achieve the proof of Lemma 9.6, one needs a little bit more analysis to estimate n≥n 0 |H n (ζ )| (and justify a posteriori the choice of |v| > 1). This also relies on (9.16), firstly noticing that the first factor in the right-hand-side can be rewritten as reaches its maximum at t 0 = c 3 v c 2 and f (t 0 ) = 1. Moreover f is increasing on (−∞, t 0 ] and decreasing on [t 0 , +∞). Integral comparison yields

By summation and the explicit calculation
one gets the upper bound where the right hand side follows directly from |v| > 1. By exactly the same computation, we get the same bound for the second factor of A(t, ζ ): for a constant C 3 depending only on B − A. By putting together the inequalities above involving C 1 , C 2 , C 3 , one obtains that ϕ satisfies the estimate (9.14) of Lemma 9.6, thus proving Theorem 9.5.
Proof. -Let ζ ∈ A,B+λ 1 such that ( ζ λ ) < 0, so that ζ / ∈ λZ. Write Now using that L − (t)dt = L + (t)dt and and summing-up the previous equalities, one obtains which makes sense whenever ( ζ λ ) = 0. For n = 0, one can make use of Cauchy's formula as in the proof of Lemma 9.6 (replacing (t) by t (t)), which leads to Similarly, one obtains for n = 1 (here the discussion depends on whether ζ ≤ B + λ 1 2 or not). Now, we proceed to estimate the remaining terms in the serie expansion giving ϕ. To this end, define for (t, ζ ) ∈ L ± × A,B+λ 1 : , Here, proceeding as for proving (9.16) and (9.17), Cauchy-Schwarz inequality gives By doing the kind of integral comparison already performed in (9.18), If we combine the previous upper-bounds, one can conclude that there exists a positive constant C 2 such that for every ζ ∈ A,B+λ 1 satisfying ( ζ λ ) ≤ −1, we get: for some constant C 4 > 0. These different estimates prove (9.19). Now, let us establish the upper-bound (9.20). For this end, observe that, on K θ , we have sin θ so that we have in particular: From Lemma 9.6, one deduces that where C = C(B − A) is the constant appearing in loc.cit. Moreover one gets trivially that |dt| |t| 2 satisfies (9.20).

9.5.3.
Version with parameters and end of the proof of Theorem 9.2. -We resume to notations introduced in Section 9.2 and the beginning of Section 9.5. We have fixed τ with τ > 0 and set Consider 0 < a < b such that a < |q|b or equivalently log a + λ 1 < log b, and let δ (a,b) > 0 small enough. For every positive number δ ≤ δ (a,b) and complex number c such that 0 < |c| ≤ δ, consider the vertical strip defined in the complex line by Consider also the subset of C 2 It is worth mentioning that B c − A c = log b − log a > λ 1 = λ does not depend on c. By a suitable choice of δ (a,b) , one can moreover assume that B c ≤ −1 − λ 1 and A c B c ≤ 2. Let (c, ζ ) be a holomorphic function defined on S a,b,δ and assume in addition that for some m ≥ 3. From the results collected in Sections 9.5.1, 9.5.2, one promptly obtains the following statement: Proof. -Define ϕ δ by the formula (compare Lemma 9.7) From this integral formula, it is clear that F − 0 and F n are well defined as holomorphic functions on S |q|a,b,δ and that m 0 depends analytically on c. Moreover, one can easily verify (as in Section 9.5.1) that the series F n converges uniformly on every compact subset of S |q|a,b,δ . Thus ϕ ∈ H(S |q|a,b,δ ) and fulfills the properties stated in the Proposition as a direct application of the construction performed in Sections 9.5.1 and 9.5.2.
Then, the proof of Theorem 9.2 immediately follows when translating this existence and uniqueness result into the original variable (z, ξ), with c = ze −2iπτ ξ and ζ = 2iπτ ξ (see Section 9.5) together with the upper bound (9.20) of Lemma 9.7.
Recall that once we have solutions to the linearized equation as achieved above, we are in position to obtain a solution to the general functional equations (9.8) and (9.7) by a standard fixed point Theorem as developed in Section 9.3. This eventually finishes (at least on sectors of opening I 1 ) the proof of Lemma A (and more precisely Lemma 4.1). One can construct others normalizing conjugacy biholomorphisms on the remaining sectors, namely I 2 , I 3 , I 4 without further real complications. This is explained below.

Construction of other sectorial normalizations.
9.6.1. On I 3 . -It is just a slight modification of the previous construction for I 1 .
The starting point consists again in the resolution of the linearized equation (9.6) on the corresponding domain. In this situation, it is relevant to deal with the levels Y = c of the first integral Y = ze 2iπτ ξ for |c| 0. One adapts the notation of Section 9.2 by defining S a,b,δ = {(z, ξ) ∈ C 2 | |Y(z, ξ)| > δ and z ∈ C a,b } (δ 0). Here again, this amounts to solve an analytic family of difference equations where the solutions satisfy some estimates which eventually leads by the same fixed point consideration to a normalizing conjugation map. This can be carried out following verbatim the same method. 9.6.2. On I 2 and I 4 . -Here, we can use the SL 2 (Z) action described in Section 1.6 to deduce existence of sectorial normalization on remaining sectors. Indeed, if we consider the cyclic covering determined by M = 0 −1 1 0 (see Section 1.6) then we are led to an alternate presentation of (U, C). Our model (U 1,0,0 , C) is now viewed as the quotient of (C * z × C ξ , {ξ = ∞}) by the semi-hyperbolic transformation In these coordinates (U, C) is isomorphic to a quotient of the form (C * z × C ξ {ξ = ∞}) by a biholomorphism F such that F (z , ∞) = (z , ∞). We can then apply the results of previous sections and obtain an analytic sectorial conjugation F • = • F 1,0,0 defined on sectorial domains of the shape arg(ξ ) ∈ I 1 =] − π, [ and arg(ξ ) ∈ I 3 = I 1 + π, = arg(τ ) = π − . Moreover, admits an asymptotic expansionˆ solving the same conjugacy equation. Finally, going back the (z, ξ) coordinates, we get that provides the sought sectorial conjugation on arg(ξ ) ∈ I 1 − =] − π, 0[= I 2 and arg(ξ ) ∈ I 3 − = I 4 taking into account that arg(ξ ) ∼ + arg(ξ ) asymptotically. Maybe composing by an automorphism of our model (U 1,0,0 , C) (see Section 3.3), we can assume that the asymptotic expansion fits with that of previous sectorial normalizations:ˆ =ˆ .  [15] under a slightly more synthesized form including the case of torsion normal bundle and finite Ueda type. As already mentioned in Section 1.2, we do not address the case where C fits into a formal fibration (corresponding to infinite Ueda type). According to a result due to Ueda, this fibration is indeed analytic and we fall into the case which satisfies the formal principal: there is no differences between analytic and formal classification and this latter is very easy to describe, see for instance [15,Section 5.1].
Let m be the torsion order of the normal bundle N C and let k > 0 be the Ueda type. Recall that this latter is necessary a multiple of m. The linear monodromy of the corresponding unitary connection along the loops 1 and τ is respectively determined by two roots of unity a 1 , a τ of respective orders m 1 and m τ such that lcm(m 1 , m τ ) = m. The triple (a 1 , a τ , k) is obviously a formal invariant of the neighborhood. A complete set of invariants is provided in [15,Section 5.2], from which we borrow and adapt the notation. In particular, there exists on each normal form described below a pencil of regular foliations (F t ) t∈P 1 fulfilling one of the following properties: 1. C is F t invariant for t ∈ C , 2. F ∞ is either tangent to C, either totally transverse to C .
Conversely every formal foliation satisfying one of these two properties fits into this pencil and is in particular convergent. Moreover, this family of foliations can be defined by a pencil of closed meromorphic forms ω t = ω 0 + tω ∞ whose expression is recalled below according to the value of the parameters characterizing the normal forms that we proceed to describe now.
Set ϕ k,ν = exp ( y k+1 1+νy k ∂ y ), and given P(z) = k −1 i=0 λ i z i a polynomial, k = mk , define (10.1) ω P := P 1 y m dy y , and g k,ν,P (y) The group Z k of k th roots of unity acts on the set of polynomials P as follows: Theorem 10.1.
- [15]Notations as above. There exist ν ∈ C and P ∈ C[z] of degree at most k − 1, unique up to the Z k -action (10.2), such that (U, C) is formally equivalent 28 to the quotient U k,ν,P,a 1 ,a τ of (C x × C y , {y = 0}) by the group generated by The pencil ω t = ω 0 + tω ∞ of closed 1-forms (and therefore foliation F t ) is generated by By duality, we get a 2-parameter family of holomorphic vector fields spanned by v 0 = τ ∂ x and v ∞ = y k+1 1 + νy k ∂ y + τ P 1 y They provide a 2-parameter group of automorphisms of the neighborhood, preserving the curve C, and the isotropy group of C is generated by v ∞ .
When m = 1, meaning that N C analytically trivial, one will denote the corresponding neighborhood simply by U k,ν,P . In that case, one easily check that φ τ is the flow at time 1 of the holomorphic vector field v k,ν,P = v 0 + v ∞ 28 As noticed in [15], this result is unchanged if one allows formal conjugacy maps inducing translation on C.

10.2.
Formal symmetries. -In order to generalize our approach to the general case with trivial normal bundle, i.e. m = 1, it is convenient to view C as a quotient of C * by a contraction: and then work in the coordinates z = e 2iπx and ξ = 1/y. The formal models described before write where F k,ν,P = exp v k,ν,P and v k,ν, , v ∞ = 2iπτ z∂ z . This can be directly borrowed from the presentation given in the (x, y) variable.
Concerning the structure of the automorphism group Aut(U k,ν,P , C) of formal automorphisms inducing translations on C, we have the following characterization which generalizes Lemma 3.9.
Lemma 10.2. -Any formal automorphism in Aut(U k,ν,P , C) is actually convergent and there exists a positive integer n = n(k, P) dividing k such that Aut(U k,ν,P , C) is isomorphic to Z/n × (C * × C * ). Moreover the Lie algebra associated to the infinitesimal action of the second factor C * × C * is spanned by v 0 and v ∞ .
Proof. -Let E be the C-vector space generated by ω 0 and ω ∞ . Set G = Aut(U k,ν,P , C). This group acts naturally on the pencil of foliations (F t ) and because there is no constant meromorphic function constant on C (finiteness of the Ueda type), also on E − {0}. Moreover, any g ∈ G acts trivially on H 1 (C), then for any ω ∈ E − {0}, ω = ω ∞ , g * ω and ω must have the same periods (one refers to [15,Section 2.4] for the notion of periods involved). As the period mapping is injective on E − {0} (see [15,Corollary 2.7]), one concludes that g * is the identity on E. In particular, according to the writing of ω 0 and ω ∞ and up to composition with the flow of v ∞ , one can assume that g restricts to identity on C and then deduce that it takes the form g(x, y) = x + yh(y), a exp ty k+1 1 + νy k ∂ y where a k = 1, ω P is invariant by the rotation y → ay and yh(y) = τ Indeed, from g |C = Id C and g * ω 0 = ω 0 , one gets g(x, y) = x + yg 1 (x, y), g 2 (y) where g 2 ∈ Diff(C, 0) is an element of the centralizer of the flow exp tv 0 at any time t ∈ C. This gives the sought expression of g 2 (cf. for instance [14][Prop. 2.17]). Now, we can exploit that g * ω ∞ = ω ∞ , which forces g 1 (x, y) = h(y) to depend only on the y variable and then to satisfy the identity d(yh(y)) = τ (g * 2 ω P − ω P ). In particular, the right hand side has to be holomorphic in the neighborhood of {y = 0}. This is the case only if ω P is invariant by the rotation y → ay.
Note that, when a = 1, g is nothing but exp tv 0 . Conversely, the transformation defined by (x, y) → (x, ay) where a satisfies the conditions above and exp tv 0 are elements of G. Set n = k if P is constant. When P has positive degree, let d P be the supremum of the positive integers d such that P is invariant under the action of the group of d th roots of unity over the polynomials (μ, P(X)) → P(μX) Set n = gcd(k, d P ). From the remarks above, we observe that for every g ∈ G, there exists (l, t) ∈ Z/n × C uniquely defined such that g = h n l • exp tv 0 and h n (z, ξ) = (z, e 2iπ n ξ ), whence the sought isomorphism. Note that n = 1 in general.
We are now ready to undertake the analytic classification.

10.3.
Trivial normal bundle, finite Ueda type: the fundamental isomorphism. -Still using the formal classification of [15] as recalled in Section 10.2, one can investigate, without any further fundamental change, the analytic classification of neighborhood of elliptic curves with arbitrary finite Ueda type. For the sake of simplicity, we will first focus on the case where the normal bundle N C of C is analytically trivial. The formal normal forms (U k,ν,P , C) are parametrized by the triple (ν, k, P) where ν is a complex number, k ∈ N >0 is the Ueda type and P is a polynomial map of one indeterminate of degree < k; P is uniquely defined modulo a certain action of the k th roots of unity on the coefficients of P as described in Section 10.2.
Let us choose 4 intervals I l i , i = 1, ..., 4 ∈ Z 4 , l = 0, ..., k − 1 ∈ Z k as follows: so that kI 0 i = I i and I 0 i ∩ I 0 i+1 = ∅, and for l = 0, set Define (see Fig. 7) where a 0 = P(0) is the constant term of P and P 1 is a polynomial of degree < k whose exact expression is not relevant in what follows. This transformation k,ν,P maps the germ of transversely sectorial domains U l i of opening I l i (which is F k,ν,P invariant) onto the germ of deleted neighborhood of L i . The full picture is thus obtained by taking the successive family of sectors with consecutive overlaps (I 0 1 , ..., I 0 4 , I 1 1 , ..........., I k−1 1 , ...., I k−1 4 ) and their corresponding system of transversal sectorial neighborhoods. This defines a cover of the deleted neighborhood U k,ν,P \ C equipped with a system of semi-local charts defined by k,ν,P . If one follows the cyclic order defined by the succession of overlapping sectors, one concludes that this deleted neighborhood can be represented by the union of deleted neighborhoods V l i − L l i of the components of a cycle D = L l i of 4k rational curves of zero type, namely the image under k,ν,P of these sectors. The neighborhood V l i of L l i is equipped with the standard coordinate system of V i (neighborhood of L i in P 1 × P 1 ).
We will also set V l i,i+1 := V l i ∩ V l i+1 if i = 0 mod 4, V l 4,1 := V l−1 4 ∩ V l 1 and consider p l i,i+1 ∈ V l i,i+1 the crossing points of D. To summarize we have thus constructed via k,ν,P a two dimensional germ of neighborhood V k,ν,P of D such that each component of D is embedded with zero self intersection and such that V k,ν,P \ D U k,ν,P \ C.
The complex structure of V k,ν,P is determined by trivial glueings (with respect to the (X, Y) coordinate) along the intersections V l i,i+1 except on V 0 4,1 where identification is made by the monodromy of k,ν,P which is given by the linear diagonal map d ν,P : Let us denote by V k,ν,P the neighborhood of four lines obtained by this process. If one specializes this series of observations to k = 1 and then P = μ automatically constant, one obtains an isomorphism between the deleted neighborhoods V 1,ν,μ \ D U 1,ν,μ \ C.
In this context, D is a cycle of 4 rational curve, and one recovers the description of Serre's example for the value (0, 0) of the parameter (ν, μ). Remark 10.3. -On V k,ν,P , the vector fields induced respectively by v ∞ and v 0 simply read as X ∞ = 2iπτ Y∂ Y and X 0 = −2iπ X∂ X − 2iπτ Y∂ Y . In particular the C * × C * part of Aut(U k,ν,P , C) corresponds at the level of V k,ν,P to transformations of type (X, Y) → (aX, bY), a, b ∈ C * . According to the description given in (the proof of) Lemma 10.2 it remains to determine the action corresponding to g = h n : (z, ξ) → (z, e 2iπ n ξ ). The latter is given in restriction to V l i , 0 ≤ l < k, by the following transformation This is done by a straightforward computation.

10.4.
The analytic moduli space. -Notations of Section 10.3. Our purpose is to give a simple characterization of neighborhoods (U, C) formally conjugated to (U k,ν,P , C) up to analytic equivalence. The way to proceed is actually suggested in the previous Section where things are basically "modeled" on Serre's example.
Let us settle some notations in accordance with Section 3. For this, denote by Diff 1 (V l i , L l i ) the group of germs of biholomorphisms of (V l i , L l i ) which preserves the germ of divisor (D ∩ V l i ) and tangent to the identity along L l i and by Diff 1 (V l i,i+1 , p l i,i+1 ) the group germs of diffeomorphisms of (V l i,i+1 , p l i,i+1 ) which preserves the germ of divisor (D ∩ V i,i+1 ) and tangent to the identity at p l i,i+1 . Analogously to the case of Serre's example, we investigate in terms of cocycle the structure of the moduli space U k,ν,P of neighborhoods (U, C) formally equivalent to (U k,ν,P , C) up to analytic equivalence. 29 10.4.1. Normalizing sectorial maps. -Concerning the construction of normalizing maps, let us explicit as before the modifications needed: let (U, C) be formally equivalent to (U k,P,ν , C).
One wants to generalize the construction performed in Section 9 and replaceˆ by a collection of transversely sectorial biholomorphisms l i ∈ G 1 (I l i ), l i =ˆ , 30 that is (10.5) F• l i = l i • F k,ν,P One adopts the notation of loc.cit without systematically mentioning the parameters (k, ν, P). As before, we will just detail the construction of a normalizing conjugation map on the germ of transversal sectorial domain determined by I 1 := I l 0 1 where l 0 ∈ Z k has been fixed. Consider the annulus C a,b = {a ≤ |z| ≤ b}, a, b > 0, b > |q| −1 a. Consider the locus where the foliation F 1 is defined by the level sets {Y k,ν,P = c} ∩ {arg ξ ∈ I 1 }, c "small" of Y k,ν,P = Y • = ze P 1 (ξ )+2iπτ ξ k k ξ 2iπτ (ν+a 0 ) = ze where R(ξ ) = o(ξ k ). It corresponds to the fibration dY = 0 on a neighborhood V l 1 of L l 1 . Let δ(a, b) > 0 sufficiently small. For every 0 < δ ≤ δ(a, b), set S a,b,δ = {(z, ξ) ∈ C 2 | |Y k,ν,P (z, ξ)| ≤ δ, z ∈ C a,b , arg ξ ∈ I 1 }.
Remark now that the conjugacy equation ( where F = • F • −1 , = • • −1 . This amounts to determine on the domain (S a,b,δ ). To do this, note that in view of the asymptotic behavior of ϕ and ϕ −1 described in Section 10.3, one has F = F 1,0,0 + ( 1, , 2, ) where one can verify that We are looking for solution of the form = Id + (g , h ). Here again, this can be reformulated as In what follows, we will indeed provide a solution of (10.8) (hence also for (10.9) by the usual transform) with "good estimates" on a domain of the form (S a,b,δ ) using a "leafwise" resolution with respect to the foliation defined by the levels of Y. This consists as before in solving a family of difference equations parametrized by the leaves space in the variable ξ where the domains are slightly modified as explained now. For every complex number c, 0 < |c| ≤ δ, consider S a,b,c = {Y k,ν,P = c} ∩ S a,b,δ and (S a,b,c ) = {(z, ξ) = (ce −2iπτ ξ , ξ)} where ξ belongs to c,a,b = ξ ∈ C : (log |c| − log b) ≤ 2iπτ ξ + Q ϕ −1 (ξ ) where a determination of the logarithm has been chosen in the sector in which we are working, namely arg ξ ∈ I 1 . Note also that Q(ϕ −1 (ξ )) = o(ξ ) and consequently the middle term in the above inequation "behaves" like (2iπτ ξ ). The remaining part of the proof then follows mutatis mutandis the same line by noticing that the equation can be then rewritten as where λ = −2iπτ , ζ = 2iπτ ξ , ϕ c (ζ ) = h(ce −ζ , ζ 2iπτ ), and c (ζ ) = δ (ce −ζ , ζ 2iπτ ) with 0 < |c| ≤ δ.
We are then reduced to solve a family difference equations in the "quasi" vertical strip c = {ζ ∈ C : log |c| − log b ≤ (ζ + Q(ϕ −1 ( ζ 2iπτ ))) ≤ log |c| − log a} depending analytically on the parameter c and we investigate the existence of a solution ϕ c on the domain c ∪ ( c + λ). This can be carried out by resolving equation (9.10) using the same method (that is, essentially Cauchy formula) replacing accordingly in the expression (9.13) the integration along the vertical lines L − , L + by l(ζ ) = A, B where l(ζ ) = (ζ − R(ϕ −1 ( ζ 2iπτ ))). One also obtain in a similar way the same kind of estimates imposing the uniqueness of the solution. This allows by the fixed point method detailed in the previous section to solve the conjugation equation under the forms (10.6) and (10.7) on the relevant domains and finally exhibit a conjugacy sectorial transformation well defined as a section of G 1 over I 1 and havingˆ as asymptotic expansion. One can analogously construct conjugacy maps on other sectors of opening I l i , i = 1, 3, l ∈ Z k and also on I l i , i = 2, 4 by exchanging the roles of the foliations F 0 and F 1 following the process described in Section 9.6.

10.4.2.
Cocycles versus analytic class and statement of the main result. -One borrows notation from Section 3 and defines G ∞ [F k,ν,P ] to be the subsheaf of G ∞ formed by germs of sectorial holomorphic transformations flat to identity alongC and commuting to F k,ν,P . By exploiting the Proposition 3.13 and the fact that (U k,ν,P , C) is "sectorially modeled" on (U 0 , C) as described in Section 10.3, one obtains the following Proposition: where φ = exp (tX 0 + t X ∞ ) has thus the form φ(X, Y) = (aX, bY) (cf. Remark 10.3). We will denote this equivalence relation by ≈.
Firstly, let us explain as in Section 3 how to associate a cocycle to a neighborhood (U, C) = (Ũ,C)/ < F > formally conjugated to (U k,ν,P , C). Recall that this means that there is a formal diffeomorphism (that can be assumed to be tangent to the identity along C)ˆ conjugating F to F k,ν,P i.e. F •ˆ =ˆ • F k,ν,P .
Recall (cf. 10.3) that for every pair (i, l), there exists a section l i of G 1 (I l i ) such that l i • F = F k,ν,P • l i . According to the description of Aut(U k,ν,P , C), namely the fact that it contains only convergent automorphisms, one can assume that the asymptotic expansion of theˆ l i is constant equal toˆ . The l i are unique up to post composition by a sectorial diffeomorphism g l i ∈ G 1 [F k,ν,P ](I l i ) having the form (10.12) where h i,l ∈ G ∞ [F k,ν,P ](I l i ). Actually, the flow of v 0 contains exactly all the automorphisms tangent to identity along C.
It follows that, on intersections I l i,i+1 = I l i ∩ I l+ε(i) i+1 , ε(i) = δ i4 , one obtains 4k germs of sectorial glueing biholomorphisms One can now invoke Proposition 10.5: setting l i := k,ν,P • l i (and taking into account the determination of k,ν,P on the corresponding sectors), one can associate 4k corresponding cocycles: except for i = 4 and l = k − 1 for which one has k−1 4 = d ν,P • ϕ 0 1,4 • 0 1 . We have therefore associated to each neighborhood (U, C) formally equivalent to (U k,ν,P , C) a cocycle ϕ = (ϕ l i,i+1 ) i∈Z 4k which is unique up to the freedom for the choice of i 's. We have now all the ingredients to state and prove our main Theorem Theorem E. -Two neighborhood (U, C) and (U , C) formally equivalent to (U k,ν,P , C) are analytically equivalent 31 if, and only if, the corresponding cocycles are equivalent (10.14) (U, C) an ∼ (U , C) ⇔ ϕ ≈ ϕ .
Moreover, every cocycle can be realized by the process described above. 31 Recall that formal/analytic conjugations are allowed to induce translations on C.
In the (z, ξ) coordinates, A corresponds to the linear map d(z, ξ) = (e 2iπβ 2 z, e 2iπ m ξ ). In particular, one has d m = id. Note that the vector field X k,ν ,P = −2iπ X∂ X on V k,ν ,P corresponds via k,ν ,P to v k ,ν ,P . Its flow at time −β 1 preserves each individual neighborhood V l i of the component L l i of the cycle and reads as (X, Y) → (e 2iπβ 1 X, Y).
The action of g v,w by "sectorial permutation" is then explicitly given in the (X, Y) coordinates (determining the neighborhood of 4k lines as described in Section 10.3) by This thus provides a "multisectorial conjugation" on eachŨ l i . One can mimic the arguments presented in Section 10 and thus obtain the description of the analytic moduli space which is thus determined by a cocycle ϕ with 4k components modulo the identifications given in Theorem E.